A higher order theory for compressible turbulent boundary layers at moderately large Reynolds number

Abstract

A higher order theory for two-dimensional turbulent boundary-layer flow of a compressible fluid past a plane wall is formulated, for moderately large values of the Reynolds number, by the method of matched asymptotic expansions. The parameters (γ − 1) M²_∞and the molecular Prandtl number are assumed to be of order unity. The analysis deals with the set of Reynolds equations of mean motion (which are underdetermined without an additional set of closure hypotheses) and assumes that the non-dimensional fluctuations in velocity, temperature and density are of orderU_*, (friction velocity divided by free-stream velocity a t some designation point), while fluctuations in pressure are of orderU²_*.The first-order results of the present study lead to asymptotic laws for velocity and temperature distributions which correspond to the law of the wall, logarithmic law and defect law, and also to skin friction and heat-transfer laws. It turns out that the first-order defect law depends upon the gradient of entropy and stagnation enthalpy and the law of the wall is independent of viscous dissipation. The second-order terms of the present work (accounting for mean convection due to turbulent mass flux, viscous dissipation in the inner flow and displacement effects in the outer flow) describe the necessary corrections to first-order terms due to low Reynolds number effects. In the overlap region the second-order results, for the law of the wall and the defect law, show bilogarithmic terms along with logarithmic terms.